1. efficient simplification of point-sampled geometry Mark Pauly Markus Gross Leif Kobbelt ETH Zurich RWTH Aachen

2. outline introduction surface model & local surface analysis point cloud simplification –hierarchical clustering –iterative simplification –particle simulation measuring surface error comparison conclusions

3. introduction 3d content creation acquisitionrenderingprocessing many applications require coarser approximations –storage –transmission –editing –rendering  surface simplification for complexity reduction

4. introduction 3d content creation acquisitionrenderingprocessing registrationraw scanspoint cloudreconstructiontriangle mesh

5. introduction 3d content creation acquisitionrenderingprocessing registrationraw scanspoint cloudreconstructiontriangle meshsimplification reduced point cloud

6. introduction 3d content creation acquisitionrenderingprocessing registrationraw scanspoint cloudsimplification reduced point cloud

7. surface model moving least squares (mls) approximation Gaussian weight function  locality idea: locally approximate surface with polynomial –compute reference plane –compute weighted least- squares fit polynomial implicit surface definition using a projection operator

8. surface model moving least squares (mls) approximation idea: locally approximate surface with polynomial –compute reference plane –compute weighted least- squares fit polynomial Gaussian weight function  locality implicit surface definition using a projection operator

9. local surface analysis local neighborhood (e.g. k-nearest)

10. local surface analysis local neighborhood (e.g. k-nearest) covariance matrix eigenproblem centroid

11. local surface analysis local neighborhood (e.g. k-nearest) eigenvectors span covariance ellipsoid surface variation smallest eigenvector is least-squares normal measures deviation from tangent plane  curvature

12. local surface analysis example originalmean curvaturevariation n=20variation n=50

13. surface simplification hierarchical clustering iterative simplification particle simulation

14. hierarchical clustering top-down approach using binary space partition recursively split the point cloud if: –size is larger than a user-specified threshold or –surface variation is above maximum threshold split plane defined by centroid and axis of greatest variation replace clusters by centroid

15. hierarchical clustering 2d example covariance ellipsoid split plane centroid root

16. hierarchical clustering 2d example

17. hierarchical clustering 2d example

18. hierarchical clustering 2d example

19. hierarchical clustering 4,280 Clusters436 Clusters43 Clusters

20. surface simplification hierarchical clustering iterative simplification particle simulation

21. iterative simplification iteratively contracts point pairs  each contraction reduces the number of points by one contractions are arranged in priority queue according to quadric error metric quadric measures cost of contraction and determines optimal position for contracted sample equivalent to QSlim except for definition of approximating planes

22. compute fundamental quadrics compute initial point-pair contraction candidates iterative simplification 2d example compute edge costs

23. iterative simplification 2d example 60.02 20.03 140.04 5 90.09 10.11 130.13 30.22 110.27 100.36 70.44 40.56 priority queue edge cost

24. iterative simplification 2d example 60.02 20.03 140.04 5 90.09 10.11 130.13 30.22 110.27 100.36 70.44 40.56 priority queue edge cost

25. iterative simplification 2d example 20.03 140.04 50.06 90.09 10.11 130.13 30.25 110.27 100.36 70.49 40.56 priority queue edge cost

26. iterative simplification 2d example 140.04 50.06 90.09 10.11 130.13 30.25 110.27 100.36 70.49 40.56 priority queue edge cost

27. iterative simplification 2d example 140.04 50.06 90.09 10.11 130.13 30.25 110.27 100.36 70.49 40.56 priority queue edge cost

28. iterative simplification 2d example 50.06 90.09 10.11 130.13 30.25 110.27 100.36 70.49 40.56 priority queue edge cost

29. iterative simplification 2d example 50.06 90.09 10.11 130.13 30.25 110.27 100.36 70.49 40.56 priority queue edge cost

30. iterative simplification 2d example 90.09 10.11 130.13 30.25 110.27 100.36 70.49 40.56 priority queue edge cost

31. iterative simplification 2d example 90.09 10.11 130.13 30.25 110.27 100.36 70.49 40.56 priority queue edge cost

32. iterative simplification 2d example 110.27 100.36 70.49 40.56 priority queue edge cost

33. iterative simplification 296,850 points2,000 points remaining contraction pairs

34. surface simplification hierarchical clustering iterative simplification particle simulation

35. resample surface by distributing particles on the surface particles move on surface according to inter- particle repelling forces particle relaxation terminates when equilibrium is reached (requires damping) can also be used for up-sampling!

36. mls surface particle simulation 2d example

37. particle simulation 2d example initialization –randomly spread particles

38. particle simulation 2d example initialization –randomly spread particles repulsion –linear repulsion force

39. projection –project particles onto surface particle simulation 2d example initialization –randomly spread particles repulsion –linear repulsion force

40. particle simulation 2d example initialization –randomly spread particles repulsion –linear repulsion force projection –project particles onto surface

41. particle simulation original model 296,850 points uniform repulsion 2,000 points adaptive repulsion 3,000 points

42. measuring error measure distance between two point-sampled surfaces S and S’ using a sampling approach compute set Q of points on S maximum error:  two-sided Hausdorff distance mean error:  area-weighted integral of point-to-surface distances size of Q determines accuracy of error measure

43. measuring error d(q,S’) measures the distance of point q to surface S’ using the mls projection operator

44. comparison: surface error error estimate for Michelangelo’s David simplified from 2,000,000 points to 5,000 points hierarchical clusteringiterative simplificationparticle simulation

45. comparison: performance execution time as a function of input model size (simplification to 1% of input model size) input size time (sec) hierarchical clustering iterative simplification particle simulation

46. comparison: performance execution time as a function of target model size (input: dragon, 435,545 points) hierarchical clustering iterative simplification particle simulation target size time (sec)

47. smoothing effect simplification up-sampling

48. point cloud vs. mesh simplification simplification  reconstruction 3.5 sec. 2.45 sec reconstruction  simplification 112.8 sec. 3.5 sec.

49. conclusions point cloud simplification can be useful to –reduce the complexity of geometric models early in the 3d content creation pipeline –build LOD surface representations –create surface hierarchies the right method depends on the application check out: www.pointshop3d.com acknowledgement: European graduate program on combinatorics, geometry, and computation